Problem 7
Prove or disprove: If there is an injection \(f: A \rightarrow B\) and a surjection \(g: A \rightarrow B\), then there is a bijection \(h: A \rightarrow B\).
Problem 8
Prove or disprove: The set \(\mathbb{Z} \times \mathbb{Q}\) is countably infinite.
Problem 8
Prove or disprove: The set \(\left\\{\left(a_{1}, a_{2}, a_{3}, \ldots\right): a_{i} \in \mathbb{Z}\right\\}\) of infinite sequences of integers is countably infinite.
Problem 9
Prove or disprove: The set \(\\{0,1\\} \times \mathbb{N}\) is countably infinite.
Problem 9
Prove that if \(A\) and \(B\) are finite sets with \(|A|=|B|,\) then any injection \(f: A \rightarrow B\) is also a surjection. Show this is not necessarily true if \(A\) and \(B\) are not finite.
Problem 9
Show that the two given sets have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula (not as a table). \(\\{0,1\\} \times \mathbb{N}\) and \(\mathbb{N}\)
Problem 10
Prove that if \(A\) and \(B\) are finite sets with \(|A|=|B|\), then any surjection \(f: A \rightarrow B\) is also an injection. Show this is not necessarily true if \(A\) and \(B\) are not finite.
Problem 10
Show that the two given sets have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula (not as a table). \(\\{0,1\\} \times \mathbb{N}\) and \(\mathbb{Z}\)
Problem 10
Prove or disprove: The set \(A=\left\\{\frac{\sqrt{2}}{n}: n \in \mathbb{N}\right\\}\) countably infinite.
Problem 11
Describe a partition of \(\mathbb{N}\) that divides \(\mathbb{N}\) into eight countably infinite subsets.
